The full covariant derivatives of the vector fields \(u^a\)  and \(e^a\)  are given by [1]}
\(\nabla _au_b&=-\mathcal {A}u_ae_b + e_ae_b\left(\frac{1}{3}\Theta + \Sigma \right) + N_{ab}\left(\frac{1}{3}\Theta -\frac{1}{2}\Sigma \right)\\&+\Omega \varepsilon _{ab},\\\nabla _ae_b&=-\mathcal {A}u_au_b + \left(\frac{1}{3}\Theta + \Sigma \right)e_au_b +\frac{1}{2}\phi N_{ab}\\&+\xi \varepsilon _{ab}.\) 
